For a particular type of elementary function, stochastic discrimination is shown to have an analytic limit function. Classifications can be performed directly by this limit function instead of by a sampling procedure. The limit function has an interpretation in terms of fields that originate from the training examples of a classification problem. Fields depend on the global configuration of the training points. The classification of a point in input space is known when the contributions of all fields are summed. Two modifications of the limit function are proposed. First, for nonlinear problems like high-dimensional parity problems, fields can be quantized. This leads to classification functions with perfect generalization for high-dimensional parity problems. Second, fields can be provided with adaptable amplitudes. The classification corresponding to a limit function is taken as an initialization; subsequently, amplitudes are adapted until an error function for the test set reaches minimal value. It is illustrated that this increases the performance of stochastic discrimination. Due to the nature of the fields, generalization improves even if the amplitude of every training example is adaptable.